Editing Mertens' theorems (section)

=== Changes in sign ===

In a paper <ref>{{Cite journal |last=Robin |first=G. |year=1983 |title=Sur l'ordre maximum de la fonction somme des diviseurs |journal=Séminaire Delange–Pisot–Poitou, Théorie des nombres (1981–1982). Progress in Mathematics|volume=38 |pages=233–244 }}</ref> on the growth rate of the [[Divisor function#Approximate growth rate|sum-of-divisors function]] published in 1983, Guy Robin proved that in Mertens' 2nd theorem the difference

:<math>\sum_{p\le n}\frac1p -\log\log n-M</math>

changes sign infinitely often, and that in Mertens' 3rd theorem the difference

:<math>\log n\prod_{p\le n}\left(1-\frac1p\right)-e^{-\gamma}</math>

changes sign infinitely often. Robin's results are analogous to [[John Edensor Littlewood|Littlewood]]'s [[Skewes's number#Skewes's numbers|famous theorem]] that the difference π(''x'')&nbsp;−&nbsp;li(''x'') changes sign infinitely often. No analog of the [[Skewes number]] (an upper bound on the first [[natural number]] ''x'' for which π(''x'')&nbsp;>&nbsp;li(''x'')) is known in the case of Mertens' 2nd and 3rd theorems.